11.1     Three-Dimensional Coordinate Systems     by SGLee-HSKim -JHLee

                                                                                                                                          http://youtu.be/_s_2T1VVob8 

 1. Draw the surface  in .

           http://matrix.skku.ac.kr/cal-lab/cal-11-1-1-a.html

     (You may do it with Sage in http://math1.skku.ac.kr/

      Refer open resources in http://math1.skku.ac.kr/pub/ )

 2. Draw the surface  in .

           http://matrix.skku.ac.kr/cal-lab/cal-11-1-2.html 

 3. Find the lengths of the sides of the triangle with vertices ,  and  .

           Is  a right triangle? Is it an isosceles triangle?

           http://matrix.skku.ac.kr/cal-lab/cal-11-1-3.html 

      Isosceles triangle because AB=BC.

 4. Find the distance from  to each of the following.

           (a) The -axis     (b) The -axis     (c) The -axis

           (d) The -plane   (e) The -plane   (f) The -plane

           http://matrix.skku.ac.kr/cal-lab/cal-11-1-4.html 

5. Find an equation of the sphere with center  and radius 3.

    What is the intersection of this sphere with the -plane?

     An equation of the sphere : ,

     and the intersection of this sphere with the -plane can be obtained by substituting  in the equation.

     Hence .

6. Find an equation of the sphere that passes through the point  and has center .

     The distance between  and  is the radius of the sphere.

      Hence, .

      Thus, an equation of sphere is .

7-8. Show that the equation represents a sphere, and find its center and radius.

7. 

       . Hence

          center: , radius: .

8. 

         Completing squares in the equation gives :

       , with the center  and radius .

9. (a) Prove that the midpoint of the line segment from

           to  is , .

    (b) Find the lengths of the medians of the triangle with vertices ,  and .

10-16. Determine the region of  represented by the equation or inequality.

10. 

     The equation  represents a plane parallel to the -plane and 8 units in front of it.

11. 

12. 

    The inequality  represents all points on or between the horizontal planes  (the -plane) and .

    So the answer is all points on or between the horizontal plane  (the -plane) and .

 13. , 

            http://matrix.skku.ac.kr/cal-lab/cal-11-1-13.html 

14. 

     The set of all points in  whose distance from the -axis is .

     This is a cylinder of radius 3 and axis along -axis.

15. 

     The inequality  is equivalent to .

     So the region consists of those points whose distance from the point  is greater than 1.

     This is the set of all points outside the sphere with radius 1 and center .

 16. 

            http://matrix.skku.ac.kr/cal-lab/cal-11-1-16.html 

17-18. Describe the given region by an inequality.

17. The half-space consisting of all points to the left of the -plane.

     This describes all points with positive -coordinates, that is, .

 18. The solid rectangular box in the first octant bounded by the planes , , and .

             http://matrix.skku.ac.kr/cal-lab/cal-11-1-18.html 

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